This paper studies the statistical characterization of ground-to-air (G2A) and reconfigurable intelligent surface (RIS)-assisted air-to-ground (A2G) communications in RIS-assisted UAV networks under the impact of channel aging. A comprehensive channel model is presented, which incorporates the time-varying fading, three-dimensional (3D) mobility, Doppler shifts, and the effects of channel aging on array antenna structures. We provide analytical expressions for the G2A signal-to-noise ratio (SNR) probability density function (PDF) and cumulative distribution function (CDF), demonstrating that the G2A SNR follows a mixture of noncentral $χ^2$ distributions. The A2G communication is characterized under RIS arbitrary phase-shift configurations, showing that the A2G SNR can be represented as the product of two correlated noncentral $χ^2$ random variables (RVs). Additionally, we present the PDF and the CDF of the product of two independently distributed noncentral $χ^2$ RVs, which accurately characterize the A2G SNR's distribution. Our paper confirms the effectiveness of RISs in mitigating channel aging effects within the coherence time. Finally, we propose an adaptive spectral efficiency method that ensures consistent system performance and satisfactory outage levels when the UAV and the ground user equipments are in motion.